1Statementofthemainresults Two-PointBoundaryValueProblemsForStronglySingularHigher-OrderLinearDifferentialEquationsWithDeviatingArguments

Electronic Journal of Qualitative Theory of Differential Equations 2012, No. 38, 1-34;http://www.math.u-szeged.hu/ejqtde/

Two-Point Boundary Value Problems For Strongly Singular Higher-Order Linear Differential Equations

With Deviating Arguments

Sulkhan Mukhigulashvili

and Nino Partsvania

Abstract

For strongly singular higher-order differential equations with deviating argu- ments, under two-point conjugated and right-focal boundary conditions, Agarwal- Kiguradze type theorems are established, which guarantee the presence of Fred- holm’s property for the above mentioned problems. Also we provide easily verifi- able best possible conditions that guarantee the existence of a unique solution of the studied problems.

2000 Mathematics Subject Classification: 34K06, 34K10

Key words and phrases: Higher order differential equation, linear, deviating argu- ment, strong singularity, Fredholm’s property.

1 Statement of the main results

1.1. Statement of the problems and the basic notations. Consider the differential equations with deviating arguments

u⁽ⁿ⁾(t) = Xm

j=1

pj(t)u^(j−1)(τj(t)) +q(t) for a < t < b, (1.1) with the two-point boundary conditions

u⁽ⁱ⁻¹⁾(a) = 0 (i= 1,· · ·, m), u^(j−1)(b) = 0 (j = 1,· · · , n−m), (1.2) u⁽ⁱ⁻¹⁾(a) = 0 (i= 1,· · · , m), u^(j−1)(b) = 0 (j =m+ 1,· · · , n). (1.3) Here n≥2, m is the integer part of n/2, −∞< a < b < +∞, pj, q ∈Lloc(]a, b[) (j = 1,· · · , m), and τj :]a, b[→]a, b[ are measurable functions. By u^(j−1)(a) (u^(j−1)(b)) we denote the right (the left) limit of the function u^(j−1) at the point a(b). Problems (1.1), (1.2), and (1.1), (1.3) are said to be singular if some or all the coefficients of (1.1) are non-integrable on [a, b], having singularities at the end-points of this segment.

∗Corresponding author.

The linear ordinary differential equations and differential equations with deviating arguments with boundary conditions (1.2) and (1.3), and with the conditions

Zb a

(s−a)ⁿ⁻¹(b−s)^2m−1[(−1)^n−mp1(s)]+ds <+∞, Zb

a

(s−a)^n−j(b−s)^2m−j|pj(s)|ds <+∞ (j = 2,· · · , m), Zb

a

(s−a)^n−m−1/2(b−s)^m−1/2|q(s)|ds <+∞,

(1.4)

and Zb

a

(s−a)ⁿ⁻¹[(−1)^n−mp1(s)]+ds <+∞, Zb

a

(s−a)^n−j|pj(s)|ds <+∞ (j = 2,· · · , m), Zb

a

(s−a)^n−m−1/2|q(s)|ds <+∞,

(1.5)

respectively, were studied by I. Kiguradze, R. P. Agarwal and some other authors (see [1], [2], [4] - [22]).

The first step in studying the linear ordinary differential equations under conditions (1.2) or (1.3), in the case when the functions pj and q have strong singularities at the points a and b, i.e. when conditions (1.4) and (1.5) are not fulfilled, was made by R. P.

Agarwal and I. Kiguradze in the article [3].

In this paper the Agarwal-Kiguradze type theorems are proved which guarantee Fred- holm’s property for problems (1.1), (1.2), and (1.1), (1.3) (see Definition 1.1). Moreover, we establish optimal, in some sense, sufficient conditions for the solvability of problems (1.1), (1.2), and (1.1), (1.3).

Throughout the paper we use the following notation.

R⁺= [0,+∞[;

[x]₊ is the positive part of numberx, that is [x]₊ = ^x+|x|₂ ;

Lloc(]a, b[) (Lloc(]a, b])) is the space of functions y:]a, b[→ R, which are integrable on [a+ε, b−ε]; ([a+ε, b]) for arbitrary small ε >0;

L_α,β(]a, b[) (L²_α,β(]a, b[)) is the space of integrable (square integrable) with the weight (t−a)^α(b−t)^β functionsy :]a, b[→R, with the norm

||y||^Lα,β = Zb

a

(s−a)^α(b−s)^β|y(s)|ds

||y||L²_α,β =Z^b

a

(s−a)^α(b−s)^βy²(s)ds1/2

; L([a, b]) =L0,0(]a, b[), L²([a, b]) =L²_0,0(]a, b[);

M(]a, b[) is the set of measurable functions τ :]a, b[→]a, b[;

Le²_α,β(]a, b[) (Le²_α(]a, b]) is the Banach space of functions y ∈ Lloc(]a, b[) (Lloc(]a, b])), satisfying

µ1 ≡maxnhZ^t

a

(s−a)^αZ^t

s

y(ξ)dξ2

dsi1/2

:a≤t≤ a+b 2

o +

+ maxnhZ^b

t

(b−s)^βZ^s

t

y(ξ)dξ2

dsi1/2

: a+b

2 ≤t≤bo

<+∞,

µ₂ ≡maxnhZ^t

a

(s−a)^αZ^t

s

y(ξ)dξ2

dsi1/2

:a≤t≤bo

<+∞. The norm in this space is defined by the equality || · ||Le²_α,β =µ1 (|| · ||Le²_α =µ2).

Ce^{n−1, m}(]a, b[) (Ce^{n−1, m}(]a, b])) is the space of functions y ∈ Ce_locⁿ⁻¹(]a, b[) (y∈Ce_locⁿ⁻¹(]a, b])), satisfying

Zb a

|y^(m)(s)|²ds <+∞. (1.6)

When problem (1.1), (1.2) is discussed, we assume that for n= 2m, the conditions pj ∈Lloc(]a, b[) (j = 1,· · · , m) (1.7) are fulfilled, and for n = 2m+ 1,along with (1.7), the conditions

lim sup

t→b

(b−t)^2m−1 Zt t1

p1(s)ds

<+∞ (t1 = a+b

2 ) (1.8)

are fulfilled. Problem (1.1), (1.3) is discussed under the assumptions

pj ∈Lloc(]a, b]) (j = 1,· · ·, m). (1.9) A solution of problem (1.1), (1.2) ((1.1), (1.3)) is sought in the space Ce^{n−1, m}(]a, b[) (Ce^{n−1, m}(]a, b])).

By h_j :]a, b[×]a, b[→R₊ and f_j :R×M(]a, b[)→C_loc(]a, b[×]a, b[) (j = 1, . . . , m) we denote the functions and the operators, respectively, defined by the equalities

h1(t, s) =

Zt s

(ξ−a)^n−2m[(−1)^n−mp1(ξ)]+dξ ,

hj(t, s) =

Zt s

(ξ−a)^n−2mpj(ξ)dξ

(j = 2,· · · , m),

(1.10)

and,

fj(c, τj)(t, s) =

Zt s

(ξ−a)^n−2m|pj(ξ)|

τZj(ξ) ξ

(ξ1−c)^2(m−j)dξ1

^1/2dξ

(j = 1,· · · , m). (1.11)

Let, moreover,

m!! =

(1 for m≤0 1·3·5· · ·m for m≥1, if m= 2k+ 1.

1.2. Fredholm type theorems.

Along with (1.1), we consider the homogeneous equation v⁽ⁿ⁾(t) =

Xm j=1

pj(t)v^(j−1)(τj(t)) for a < t < b. (1.10) In the case where conditions (1.4) and (1.5) are violated, the question on the presence of the Fredholm’s property for problem (1.1), (1.2) ((1.1), (1.3)) in some subspace of the space Ce_loc^n−1,m(]a, b[) (Ce_loc^n−1,m(]a, b])) remains so far open. This question is answered in Theorem 1.1 (Theorem 1.2 ) formulated below which contains optimal in a certain sense conditions guaranteeing the Fredholm’s property for problem (1.1), (1.2) ((1.1), (1.3)) in the space Ce^{n−1, m}(]a, b[) (Ce^{n−1, m}(]a, b])).

Definition 1.1. We will say that problem (1.1), (1.2) ((1.1), (1.3)) has the Fredholm’s property in the space Ce^n−1,m(]a, b[) (Ce^n−1,m(]a, b])), if the unique solvability of the cor- responding homogeneous problem (1.10), (1.2) ((1.10), (1.3)) in that space implies the unique solvability of problem (1.1), (1.2) ((1.1), (1.3)) for every q∈Le²_{2n−2m−2,2m−2}(]a, b[) (q∈Le²_2n−2m−2(]a, b])).

Theorem 1.1. Let there exist a₀ ∈]a, b[, b₀ ∈]a₀, b[, numbers l_kj > 0, γ_kj > 0, and functions τj ∈M(]a, b[) (k = 0,1, j = 1, . . . , m) such that

(t−a)^2m−jhj(t, s)≤l0j for a < t≤s≤a0, lim sup

t→a

(t−a)^{m−12−γ0j}f_j(a, τ_j)(t, s)<+∞, (1.12) (b−t)^2m−jhj(t, s)≤l1j for b0 ≤s ≤t < b,

lim sup

t→b

(b−t)^{m−12−γ1j}fj(b, τj)(t, s)<+∞, (1.13)

and Xm

j=1

(2m−j)2^2m−j+1

(2m−1)!!(2m−2j+ 1)!! lkj <1 (k= 0,1). (1.14)

Let, moreover,(1.10), (1.2)have only the trivial solution in the spaceCe^n−1,m(]a, b[). Then problem (1.1), (1.2) has the unique solution u for every q ∈ Le²_{2n−2m−2,2m−2}(]a, b[), and there exists a constant r, independent of q, such that

||u^(m)||L² ≤r||q||Le²_{2n−2m−2,2m−2}. (1.15) Corollary 1.1. Let numbers κ_kj, ν_kj ∈R⁺ be such that

νk1 >2n+ 2−2k(2m−n), νkj >2 (k = 0,1; j = 2, . . . , m), (1.16) lim sup

t→a

|τj(t)−t|

(t−a)^ν0j <+∞, lim sup

t→b

|τj(t)−t|

(b−t)^ν1j <+∞, (1.17)

and Xm

j=1

2^2m−j+1

(2m−1)!!(2m−2j+ 1)!!κkj <1 (k = 0,1). (1.18) Moreover, let κ∈R⁺, p_0j ∈L_n−j,2m−j(]a, b[;R⁺), and

− κ

[(t−a)(b−t)]²ⁿ −p01(t)≤(−1)^n−mp1(t)≤

≤ κ₀₁

(t−a)ⁿ + κ₁₁

(t−a)^n−2m(b−t)^2m +p₀₁(t), (1.19)

|pj(t)| ≤ κ0j

(t−a)^n−j+1 + κ1j

(t−a)^n−2m(b−t)^2m−j+1 +p0j(t) (j = 2, . . . , m). (1.20) Let, moreover,(1.10), (1.2)have only the trivial solution in the spaceCe^n−1,m(]a, b[). Then problem (1.1), (1.2) has the unique solution u for every q ∈ Le²_{2n−2m−2,2m−2}(]a, b[), and there exists a constant r, independent of q, such that (1.15) holds.

Theorem 1.2. Let there exist a0 ∈]a, b[, numbers l0j > 0, γ0j > 0, and functions τj ∈ M(]a, b[) such that condition (1.12) is fulfilled and

Xm j=1

(2m−j)2^2m−j+1

(2m−1)!!(2m−2j+ 1)!!l0j <1. (1.21) Let, moreover, problem (1.1₀), (1.3)have only the trivial solution in the spaceCe^n−1,m(]a, b]).

Then problem (1.1), (1.3) has the unique solution u for every q ∈ Le²_2n−2m−2(]a, b]), and there exists a constant r, independent of q, such that

||u^(m)||L² ≤r||q||Le²_2n−2m−2. (1.22) Corollary 1.2. Let numbers κ0j, ν0j ∈R⁺ be such that

ν01>2n+ 2, ν0j ≥2 (j = 2, . . . , m), (1.23) lim sup

t→a

|τj(t)−t|

(t−a)^ν0j <+∞, (1.24)

and _m X

j=1

2^2m−j+1

(2m−1)!!(2m−2j+ 1)!!κ_0j <1. (1.25) Let, moreover, κ∈R⁺, p0j ∈Ln−j,0(]a, b];R⁺), and

− κ

(t−a)²ⁿ −p01(t)≤(−1)^n−mp1(t)≤ κ₀₁

(t−a)ⁿ +p01(t), (1.26)

|pj(t)| ≤ κ0j

(t−a)^n−j+1 +p0j(t) (j = 2, . . . , m). (1.27) Let, moreover, problem (1.10), (1.3)have only the trivial solution in the spaceCe^n−1,m(]a, b]).

Then problem (1.1), (1.3) has the unique solution u for every q ∈ Le²_2n−2m−2(]a, b]), and there exists a constant r, independent of q, such that (1.22) holds.

Theorem 1.3. Let c1 =a, c2 =b, ess sup

a<t<b

1

|t−c_i|^m+1−j

τj(t)

Z

t

|ξ−ci|^m−j−1dξ

<+∞ (j = 1, . . . , m) (1.28)

if i= 1,2 (if i= 1),

pj ∈Ln−j,2m−j(]a, b[)

pj ∈Ln−j,0(]a, b])

(j = 1, . . . , m), (1.29) and let problem (1.1), (1.2) ((1.1), (1.3))be uniquely solvable in the space Ce^{n−1, m}(]a, b[) (in the spaceCe^{n−1, m}(]a, b]). Then this problem is uniquely solvable in the spaceCeⁿ⁻¹(]a, b[) (in the space Ceⁿ⁻¹(]a, b]) as well.

Remark 1.1. In [3], an example is constructed which demonstrates that if condition (1.29) is violated, then problem (1.1), (1.2) (problem (1.1), (1.3)) with τj(t) ≡ t(j = 1, . . . , m) may be uniquely solvable in the spaceCe^n−1,m(]a, b[) (in the spaceCe^n−1,m(]a, b])) and this problem may have infinite set of solutions in the space Ce^loc(]a, b[) (in the space Ce^loc(]a, b])).

Also, in [3] it is demonstrated that strict inequalities (1.14), (1.21), (1.18), (1.25) are sharp because they cannot be replaced by nonstrict ones.

1.2. Existence and uniqueness theorems.

Theorem 1.4. Let there exist numbers t^∗ ∈]a, b[, ℓkj > 0, lkj ≥ 0, and γkj > 0 (k = 0,1;j = 1, . . . , m) such that along with

Xm j=1

(2m−j)2^2m−j+1l0j

(2m−1)!!(2m−2j+ 1)!! + 2^2m−j−1(t^∗−a)^γ0jl0j

(2m−2j−1)!!(2m−3)!!p 2γ_0j

< 1

2, (1.30) Xm

j=1

(2m−j)2^2m−j+1l1j

(2m−1)!!(2m−2j+ 1)!! + 2^2m−j−1(b−t^∗)^γ0jl1j

(2m−2j−1)!!(2m−3)!!p 2γ1j

< 1

2, (1.31)

the conditions

(t−a)^2m−jhj(t, s)≤l0j, (t−a)^{m−γ0j−1/2}fj(a, τj)(t, s)≤l0j for a < t≤s≤t^∗, (1.32) (b−t)^2m−jhj(t, s)≤l1j, (b−t)^{m−γ1j−1/2}fj(b, τj)(t, s)≤l1j for t^∗≤s≤t < b (1.33) hold. Then for everyq ∈Le²_{2n−2m−2,2m−2}(]a, b[)problem (1.1), (1.2) is uniquely solvable in the space Ce^{n−1, m}(]a, b[).

To illustrate this theorem, we consider the second order differential equation with a deviating argument

u^′′(t) =p(t)u(τ(t)) +q(t), (1.34) under the boundary conditions

u(a) = 0, u(b) = 0. (1.35)

From Theorem 1.4, with n = 2, m = 1, t^∗ = (a+b)/2, γ01 =γ11 = 1/2, l01 = l11 = κ0, l01=l11 =√

2κ1/√

b−a, we get

Corollary 1.3. Let function τ ∈M(]a, b[) be such that 0≤τ(t)−t≤ 2⁶

(b−a)⁶(t−a)⁷ for a < t≤ a+b 2 ,

− 2⁶

(b−a)⁶(b−t)⁷ ≤t−τ(t)≤0 for a+b

2 ≤t < b.

(1.36)

Moreover, let function p:]a, b[→R and constants κ0, κ1 be such that

− 2⁻²(b−a)²κ0

[(b−t)(t−a)]² ≤p₁(t)≤ 2⁻⁷(b−a)⁶κ1

[(b−t)(t−a)]⁴ for a < t≤b (1.37) and

4κ0 +κ1 < 1

2. (1.38)

Then for every q ∈ Le²_0,0(]a, b[) problem (1.34), (1.35) is uniquely solvable in the space Ce^1,1(]a, b[).

Theorem 1.5. Let there exist numbers t^∗ ∈]a, b[, ℓ0j > 0, ℓ0j ≥ 0, and γ0j > 0 (j = 1, . . . , m) such that conditions

(t−a)^2m−jhj(t, s)≤l0j, (t−a)^{m−γ0j−1/2}fj(a, τj)(t, s)≤l0j for a < t≤s≤b, (1.39) and

Xm j=1

(2m−j)2^2m−j+1l_0j

(2m−1)!!(2m−2j+ 1)!! + 2^2m−j−1(t^∗−a)^γ0jl_0j (2m−2j−1)!!(2m−3)!!p

2γ0j

<1 (1.40) hold. Then for every q ∈Le²_2n−2m−2(]a, b]) problem (1.1), (1.3) is uniquely solvable in the space Ce^{n−1, m}(]a, b]).

Theorem 1.6. Let there exist numbers t^∗ ∈]a, b[, ℓkj > 0, lkj ≥ 0, and γkj > 0 (k = 0,1;j = 1, . . . , m) such that along with (1.40) and

Xm j=1

(2m−j)2^2m−j+1l_1j

(2m−1)!!(2m−2j + 1)!!+ 2^2m−j−1(b−t^∗)^γ0jl_1j (2m−2j−1)!!(2m−3)!!p

2γ1j

<1, (1.41) conditions (1.32), (1.33) hold. Moreover, let τj ∈M(]a, b[) (j = 1, . . . , n) and

sign[(τj(t)−t^∗)(t−t^∗)]≥0 for a < t < b. (1.42) Then for every q ∈ Le²_{2n−2m−2,2m−2}(]a, b[) problem (1.1), (1.2) is uniquely solvable in the space Ce^{n−1, m}(]a, b[).

Also, from Theorem 1.6, with n = 2, m = 1, t^∗ = (a+b)/2, γ₀₁ = γ₁₁ = 1/2, l₀₁ = l11 =κ0, l01=l11=√

2κ1/√

b−a, we get

Corollary 1.4. Let functions p:]a, b[→R, τ ∈M(]a, b[) and constants κ0 >0, κ1 >0 be such that along with (1.36) and (1.37) the inequalities

sign[(τ(t)− a+b

2 )(t−a+b

2 )]≥0 for a < t < b (1.43) and

4κ₀+κ₁ <1 (1.44)

hold. Then for every q ∈ Le²_0,0(]a, b[) problem (1.34), (1.35) is uniquely solvable in the space Ce^1,1(]a, b[).

2 Auxiliary propositions

2.1. Lemmas on integral inequalities. Now we formulate two lemmas which are proved in [3].

Lemma 2.1. Let ∈Ce_loc^m−1(]t₀, t₁[) and

u^(j−1)(t0) = 0 (j = 1, . . . , m),

t1

Z

t0

|u^(m)(s)|²ds <+∞. (2.1)

Then Zt t0

(u^(j−1)(s))²

(s−t0)^2m−2j+2ds≤ 2^m−j+1 (2m−2j+ 1)!!

2Zt t0

|u^(m)(s)|²ds for t0 ≤t≤t1. (2.2)

Lemma 2.2. Let u∈Ce_loc^m−1(]t0, t1[), and u^(j−1)(t1) = 0 (j = 1, . . . , m),

t1

Z

t0

|u^(m)(s)|²ds <+∞. (2.3) Then

t1

Z

t

(u^(j−1)(s))²

(t1−s)^2m−2j+2ds≤ 2^m−j+1 (2m−2j+ 1)!!

2Zt1

t

|u^(m)(s)|²ds for t₀ ≤t≤t₁. (2.4) Let t0, t1 ∈]a, b[, u∈Ce_loc^m−1(]t0, t1[) and τj ∈M(]a, b[) (j = 1, . . . , m).Then we define the functions µj : [a, (a+b)/2]×[(a+b)/2, b]×[a, b]→ [a, b], ρk : [t0, t1]→ R+(k = 0,1), λ_j : [a, b]×]a, (a+b)/2]×[(a+b)/2, b[×]a, b[→R₊, by the equalities

µj(t0, t1, t) =







τj(t) for τj(t)∈[t0, t1] t0 for τj(t)< t0

t1 for τj(t)> t1

,

ρ_k(t) =

t_k

Z

t

|u^(m)(s)|²ds

, λ_j(c, t₀, t₁, t) =

µj(tZ0,t1,t) t

(s−c)^2(m−j)ds

1/2.

(2.5)

Let also functions αj : R₊³ ×[0,1[→ R+ and βj ∈ R+×[0,1[→ R+ (j = 1,· · · , m) be defined by the equalities

αj(x, y, z, γ) =x+ 2^m−jy z^γ

(2m−2j−1)!!, βj(y, γ) = 2^2m−j−1

(2m−2j−1)!!(2m−3)!!

y^γ

√2γ. (2.6) Lemma 2.3. Let a0 ∈]a, b[, t0 ∈]a, a0[, t1 ∈]a0, b[, and the function u ∈ Ce_loc^m−1(]t0, t1[) be such that conditions (2.1) hold. Moreover, let constants l0j >0, l0j ≥0, γ0j >0, and functions p_j ∈L_loc(]t₀, t₁[), τ_j ∈M(]a, b[) be such that the inequalities

(t−t0)^2m−1

a0

Z

t

[p₁(s)]+ds≤l0 1, (2.7)

(t−t₀)^2m−j

a0

Z

t

p_j(s)ds

≤l_0j (j = 2, . . . , m), (2.8)

(t−t0)^{m−12−γ0j}

a0

Z

t

p_j(s)λj(t0, t0, t1, s)ds

≤l0j (j = 1, . . . , m) (2.9) hold for t0 < t≤a0. Then

a0

Z

t

p_j(s)u(s)u^(j−1)(µj(t0, t1, s))ds≤

≤αj(l0j, l0j, a0−a, γ0j)ρ^1/2₀ (τ^∗)ρ^1/2₀ (t) +l0jβj(a0−a, γ0j)ρ^1/2₀ (τ^∗)ρ^1/2₀ (a0)+

+l0j

(2m−j)2^2m−j+1

(2m−1)!!(2m−2j+ 1)!!ρ0(a0) for t0 < t≤a0,

(2.10)

where τ^∗ = sup{µj(t0, t1, t) :t0 ≤t ≤a0, j = 1, . . . , m} ≤t1.

Proof. In view of the formula of integration by parts, fort ∈[t₀, a₀] we have

a0

Z

t

p_j(s)u(s)u^(j−1)(µj(t0, t1, s))ds=

a0

Z

t

p_j(s)u(s)u^(j−1)(s)ds+

+

a0

Z

t

p_j(s)u(s) ^µj(tZ^0,t1,s)

s

u^(j)(ξ)dξ

ds =u(t)u^(j−1)(t)

a0

Z

t

p_j(s)ds+

+ X1 k=0

a0

Z

t

Z^a0

s

p_j(ξ)dξ

u^(k)(s)u^(j−k)(s)ds+

a0

Z

t

p_j(s)u(s) ^µj(tZ^0,t1,s)

s

u^(j)(ξ)dξ ds

(2.11)

(j = 2, . . . , m), and

a0

Z

t

p₁(s)u(s)u(µ1(t0, t1, s))ds≤

a0

Z

t

[p₁(s)]+u²(s)ds+

+

a0

Z

t

|p₁(s)u(s)|

µ1(tZ0,t1,s) s

u^′(ξ)dξ

ds≤u²(t)

a0

Z

t

[p₁(s)]+ds+

+2

a0

Z

t

Z^a0

s

[p₁(ξ)]+dξ

|u(s)u^′(s)|ds+

a0

Z

t

|p₁(s)u(s)|

µ1(tZ0,t1,s) s

u^′(ξ)dξ ds.

(2.12)

On the other hand, by conditions (2.1), the Schwartz inequality and Lemma 2.1, we deduce that

|u^(j−1)(t)|= 1 (m−j)!

Zt t0

(t−s)^m−ju^(m)(s)ds

≤(t−t0)^m−j+1/2ρ^1/2₀ (t) (2.13) for t₀ ≤t≤a₀ (j = 1, . . . , m).If along with this, in the case j > 1, we take into account inequality (2.8), and lemma 2.1, for t ∈[t0, a0], we obtain the estimates

u(t)u^(j−1)(t)

a0

Z

t

p_j(s)ds

≤(t−t₀)^2m−j

a0

Z

t

p_j(s)ds

ρ₀(t)≤l_0jρ₀(t), (2.14)

and X1 k=0

a0

Z

t

Z^a0

s

p_j(ξ)dξ

u^(k)(s)u^(j−k)(s)ds ≤l0j

X1 k=0

a0

Z

t

|u^(k)(s)u^(j−k)(s)| (s−t0)^2m−j ds≤

≤ l0j

X1 k=0

Z^a0

t

|u^(k)(s)|²ds (s−t0)^2m−2k

1/2Z^a0

t

|u^(j−k)(s)|²ds (s−t0)^2m+2k−2j

1/2

≤

≤l0jρ0(a0) X1

k=0

2^2m−j

(2m−2k−1)!!(2m+ 2k−2j−1)!!.

(2.15)

Analogously, if j = 1, by (2.7) we obtain u²(t)

a0

Z

t

[p₁(s)]₊ds ≤l₀₁ρ₀(t),

2

a0

Z

t

Z^a0

s

[p₁(ξ)]₊dξ

|u(s)u^′(s)|ds≤l₀₁ρ₀(a₀)(2m−1)2^2m [(2m−1)!!]²

(2.16)

for t0 < t≤a0.

By the Schwartz inequality, Lemma 2.1, and the fact thatρ0 is nondecreasing function, we get

µj(tZ0,t1,s) s

u^(j)(ξ)dξ

≤ 2^m−j

(2m−2j−1)!!λj(t0, t0, t1, s)ρ^1/2₀ (τ^∗) (2.17) for t0 < s≤a0. Also, due to (2.2), (2.9) and (2.13), we have

|u(t)|

a0

Z

t

|p_j(s)|λj(t0, t0, t1, s)ds= (t−t0)^m−1/2ρ^1/2₀ (t)

a0

Z

t

|p_j(s)|λj(t0, t0, t1, s)ds≤

≤l_0j(t−t₀)^γ0jρ^1/2₀ (t),

a0

Z

t

|u^′(s)|Z^a0

s

|p_j(ξ)|λj(t0, t0, t1, ξ)dξ

ds ≤l0j a0

Z

t

|u^′(s)|

(s−t0)^{m−12−γ0j}ds≤

≤l0j

2^m−1(a0−a)^γ0j (2m−3)!!p

2γ0j

ρ^1/2₀ (a0) for t0 < t≤a0.From the last three inequalities it is clear that

(2m−2j −1)!!

2^m−jρ^1/2₀ (τ^∗)

a0

Z

t

p_j(s)u(s) ^µj(tZ^0,t1,s)

s

u^(j)(ξ)dξ ds

≤

a0

Z

t

|p_j(s)u(s)|λj(t0, t0, t1, s)ds≤

≤ |u(t)|

a0

Z

t

|p_j(s)|λj(t0, t0, t1, s)ds+

a0

Z

t

|u^′(s)|Z^a0

s

|p_j(ξ)|λj(t0, t0, t1, ξ)dξ ds≤

≤l0j(t−t0)^γ0jρ^1/2₀ (t) +l0j

2^m−1(a₀−a)^γ0j (2m−3)!!p

2γ0j

ρ^1/2₀ (a0)

(2.18)

for t0 < t ≤ a0. Now, note that from (2.11) and (2.12) by (2.14)-(2.16) and (2.18), it immediately follows inequality (2.10).

The following lemma can be proved similarly to Lemma 2.3.

Lemma 2.4. Let b0 ∈]a, b[, t1 ∈]b0, b[, t0 ∈]a, b0[, and the function u∈Ce_loc^m−1(]t0, t1[) be such that conditions (2.3) hold. Moreover, let constants l1j > 0, l1j ≥ 0, γ1j > 0, and functions p_j ∈Lloc(]t0, t1[), τj ∈M(]a, b[) be such that the inequalities

(t1−t)^2m−1 Zt b0

[p₁(s)]+ds≤l1 1, (2.19)

(t1−t)^2m−j

Zt b0

p_j(s)ds

≤l1j (j = 2, . . . , m), (2.20)

(t₁−t)^{m−12−γ1j}

Zt b0

p_j(s)λ_j(t₁, t₀, t₁, s)ds

≤l_1j (j = 1, . . . , m) (2.21)

hold for b0 < t≤t1. Then Zt b0

p_j(s)u(s)u^(j−1)(µj(t0, t1, s))ds≤

≤α_j(l_1j, l_1j, b−b₀, γ_1j)ρ^1/2₁ (τ_∗)ρ^1/2₁ (t) +l_1jβ_j(b−b₀, γ_1j)ρ^1/2₁ (τ_∗)ρ^1/2₁ (b₀)+

+l1j

(2m−j)2^2m−j+1

(2m−1)!!(2m−2j+ 1)!!ρ1(b0) for b0 ≤t < t1,

(2.22)

where τ∗ = inf{µj(t0, t1, t) :b0 ≤t≤t1, j = 1, . . . , m} ≥t0.

2.2. Lemma on the property of functions from the space Ce^n−1,m(]a, b[).

Lemma 2.5. Let

w(t) =

n−mX

i=1 n−mX

k=i

cik(t)u^(n−k)(t)u⁽ⁱ⁻¹⁾(t),

where Ce^n−1,m(]a, b[), and each cik : [a, b] → R is an (n−k −i+ 1) -times continuously differentiable function. Moreover, if

u⁽ⁱ⁻¹⁾(a) = 0 (i= 1, . . . , m), lim sup

t→a

|cii(t)|

(t−a)^n−2m <+∞ (i= 1, . . . , n−m), then

lim inf

t→a |w(t)|= 0, and if u⁽ⁱ⁻¹⁾(b) = 0 (i= 1, . . . , n−m), then

lim inf

t→b |w(t)|= 0.

The proof of this lemma is given in [9].

2.3. Lemmas on the sequences of solutions of auxiliary problems.

Now for every natural k we consider the auxiliary boundary problems u⁽ⁿ⁾(t) =

Xm j=1

pj(t)u^(j−1)(µj(t0k, t1k, t)) +qk(t) for t0k ≤t≤t1k, (2.23) u⁽ⁱ⁻¹⁾(t0k) = 0 (i= 1, . . . , m), u^(j−1)(t1k) = 0 (j = 1, . . . , n−m), (2.24) where

a < t0k < t1k < b (k∈N), lim

k→+∞t0k =a, lim

k→+∞t1k=b, (2.25) and

u⁽ⁿ⁾(t) = Xm

j=1

pj(t)u^(j−1)(µj(t0k, b, t)) +qk(t) for t0k ≤t≤b, (2.26) u⁽ⁱ⁻¹⁾(t0k) = 0 (i= 1, . . . , m), u^(j−1)(b) = 0 (j = 1, . . . , n−m), (2.27) where

a < t0k< b (k ∈N), lim

k→+∞t0k =a. (2.28)

Throughout this section, when problems (1.1), (1.2) and (2.23), (2.24) are discussed we assume that

pj ∈Lloc(]a, b[) (j = 1, ..., m), q, qk∈Le²2n−2m−2,2m−2(]a, b[), (2.29) and for an arbitrary (m−1)-times continuously differentiable function x :]a, b[→ R, we set

Λk(x)(t) = Xm

j=1

pj(t)x^(j−1)(µj(t0k, t1k, t)), Λ(x)(t) = Xm

j=1

pj(t)x^(j−1)(τj(t)). (2.30) Problems (1.1), (1.3) and (2.26), (2.27) are considered in the case

pj ∈Lloc(]a, b]) (j = 1, ..., m), q, qk ∈Le²_{2n−2m−2,0}(]a, b[), (2.31) and for an arbitrary (m−1)-times continuously differentiable function x:]a, b] →R, we set

Λk(x)(t) = Xm

j=1

pj(t)x^(j−1)(µj(t0k, b, t)), Λ(x)(t) = Xm

j=1

pj(t)x^(j−1)(τj(t)). (2.32) Remark 2.1. From the definition of the functions µj (j = 1, . . . , m), the estimate

|µj(t0k, t1k, t)−τj(t)| ≤

(0 for τj(t)∈]t0k, t1k[ max{b−t1k, t0k−a} for τj(t)6∈]t0k, t1k[ follows. Thus, if conditions (2.25) hold, then

k→+∞lim µ_j(t_0k, t_1k, t) =τ_j(t) (j = 1, . . . , m) uniformly in ]a, b[. (2.33)

Lemma 2.6. Let conditions (2.25)hold and the sequence of the(m−1)-times continuously differentiable functions xk :]t0k, t1k[→ R, and functions x^(j−1) ∈ C([a, b]) (j = 1, . . . , m) be such that

k→+∞lim x^(j−1)_k (t) =x^(j−1)(t) (j = 1, . . . , m) uniformly in ]a, b[ (]a, b]). (2.34) Then for any nonnegative function w∈C([a, b]) and t^∗ ∈]a, b[,

k→+∞lim Zt t^∗

w(s)Λk(xk)(s)ds= Zt t^∗

w(s)Λ(x)(s)ds (2.35)

uniformly in ]a, b[, where Λk and Λ are defined by equalities (2.30).

Proof. We have to prove that for any δ∈]0, min{b−t^∗, t^∗−a}[,and ε >0, there exists a constant n0 ∈N such that

Zt t^∗

w(s)(Λk(xk)(s)−Λ(x)(s))ds

≤ε for t∈[a+δ, b−δ], k > n0. (2.36)

Let, now w(t∗) = max

a≤t≤bw(t), and ε1 = ε 2w(t∗)

Pm j=1

Rb−δ

a+δ |pj(s)|ds−1

. Then from the inclusions x^(j−1)_k ∈ C([a+δ, b−δ]), x^(j−1) ∈ C([a, b]) (j = 1, . . . , m), conditions (2.33) and (2.34), it follows the existence of such constant n0 ∈N that

|x^(j−1)_k (µj(t0k, t1k, s))−x^(j−1)(µj(t0k, t1k, s))| ≤ε1,|x^(j−1)(µj(t0k, t1k, s))−x^(j−1)(τj(s))| ≤ε1

for t∈[a+δ, b−δ], k > n₀, j = 1, . . . , m.Thus from the inequality

|Λk(xk)(s)−Λ(x)(s)| ≤ |Λk(xk)(s)−Λk(x)(s)|+|Λk(x)(s)−Λ(x)(s)| ≤2ε1

Xm j=1

|pj(t)|, we have (2.36).

The proof of the following lemma is analogous to that of Lemma 2.6.

Lemma 2.7. Let conditions (2.28)hold and the sequence of the(m−1)-times continuously differentiable functions xk :]t0k, b] → R, and functions x^(j−1) ∈ C([a, b]) (j = 1, . . . , m) be such that lim

l→+∞x^(j−1)_k (t) = x^(j−1)(t) (j = 1, . . . , m) uniformly in ]a, b]. Then for any nonnegative function w∈C([a, b]), andt^∗ ∈]a, b], condition (2.35) holds uniformly in ]a, b], where Λk and Λ are defined by equalities (2.32).

Lemma 2.8. Let condition (2.25) hold, and for every natural k, problem (2.23), (2.24) have a solution uk ∈Ce_locⁿ⁻¹(]a, b[), and there exist a constant r0 >0 such that

t1k

Z

t_0k

|u^(m)_k (s)|ds ≤r₀² (k ∈N) (2.37)

holds, and if n = 2m+ 1, let there exist constants ρj ≥0, ρ_j ≥0, γ1j >0 such that ρj = supn

(b−t)^2m−j

Zt t1

(s−a)pj(s)ds

:t0 ≤t < bo

<+∞,

ρ_j = supn

(b−t)^{m−γ1j−1/2} Zt t1

(s−a) p_j(s)

λ_j(b, t_0k, t_1k, s)ds:t₀ ≤t < bo

<+∞,

(2.38)

for t1 = ^a+b₂ , (j = 1, ..., m). Moreover, let

k→+∞lim ||qk−q||Le²_{2n−2m−2,2m−2} = 0, (2.39) and the homogeneous problem (1.1₀), (1.2) have only the trivial solution in the space Ce^n−1,m(]a, b[). Then nonhomogeneous problem (1.1), (1.2) has a unique solution u such that

||u^(m)||L² ≤r₀, (2.40)

and

k→+∞lim u^(j−1)_k (t) = u^(j−1)(t) (j = 1, . . . , n) uniformly in ]a, b[ (2.41) (that is, uniformly on [a+δ, b−δ] for an arbitrarily small δ >0).

Proof. Suppose t1, . . . , tn are the numbers such that a+b

2 =t1 <· · ·< tn < b, (2.42) and gi(t) are the polynomials of (n−1)-th degree, satisfying the conditions

gj(tj) = 1, gj(ti) = 0 (i6=j; i, j = 1, . . . , n). (2.43) Then for every natural k,for the solution uk of problem (2.23), (2.24) the representation

u_k(t) = Xn

j=1

u_k(t_j)− 1 (n−1)!

tj

Z

t1

(t_j−s)ⁿ⁻¹(Λ_k(u_k)(s) +q_k(s))ds g_j(t)+

+ 1

(n−1)!

Zt t1

(t−s)ⁿ⁻¹(Λk(uk)(s) +qk(s))ds (2.44) is valid. For an arbitraryδ ∈]0, ^a+b₂ [, we have

t1

Z

t

(s−t)^n−j(qk(s)−q(s))ds

= (n−j)

t1

Z

t

(s−t)^n−j−1Z^t1

s

(qk(ξ)−q(ξ))dξ ds

≤

≤nZ^t1

t

(s−a)^2m−2jds1/2Z^t1

t

(s−a)^2n−2m−2Z^t1

s

(qk(ξ)−q(ξ))dξ2

ds1/2

≤

≤n

(t₁−a)^2m−2j+1−δ^2m−2j+1

1/2||q_k−q||Le²_{2n−2m−2,2m−2} for a+δ ≤t≤t₁,

Zt t1

(t−s)^n−j(qk(s)−q(s))ds ≤n

(b−t1)^{2n−2m−2j+1}−δ^{2n−2m−2j+1}

1/2×

×||qk−q||Le²_{2n−2m−2,2m−2} for t1 ≤t≤ b−δ (j = 1, . . . , n−1).

(2.45)

Hence, by condition (2.39), we find

k→+∞lim

t1

Z

t

(s−t)^n−j(qk(s)−q(s))ds= 0 uniformly in ]a, b[ (j = 1, . . . , n−1). (2.46) Analogously one can show that if t0 ∈]a, b[, then

k→+∞lim Zt t0

(s−t0)(qk(s)−q(s))ds= 0 uniformly on I(t0), (2.47) whereI(t0) = [t0,(a+b)/2] fort0 <(a+b)/2 and I(t0) = [(a+b)/2, t0] fort0 >(a+b)/2.

In view of inequalities (2.37), the identities u^(j−1)_k (t) = 1

(m−j)!

Zt tik

(t−s)^m−ju^(m)_k (s)ds (2.48) for i= 0,1; j = 1, . . . , m; k ∈N,yield

|u^(j−1)_k (t)| ≤rj[(t−a)(b−t)]^m−j+1/2 (2.49) for t0k ≤t≤t1k (j = 1, . . . , m; k∈N), where

rj = r₀

(m−j)!(2m−2j+ 1)^−1/2 2 b−a

m−j+1/2

(j = 1, . . . , m). (2.50) By virtue of the Arzela-Ascoli lemma and conditions (2.37) and (2.49), the sequence {uk}^+∞k=1 contains a subsequence {uk_l}^+∞l=1 such that {u^(j−1)_kl }^+∞l=1 (j = 1, . . . , m) are uni- formly convergent in ]a, b[. Suppose

l→+∞lim uk_l(t) = u(t). (2.51)

Then in view of (2.49), u^(j−1) ∈C([a, b]) (j = 1, . . . , m), and

l→+∞lim u^(j−1)_kl (t) =u^(j−1)(t) (j = 1, . . . , m) uniformly in ]a, b[. (2.52) If along with this we take into account conditions (2.25) and (2.46), from (2.44) by lemma 2.6 we find

u(t) = Xn

j=1

u(tj)− 1 (n−1)!

tj

Z

t1

(tj −s)ⁿ⁻¹(Λ(u)(s) +q(s))ds gj(t)+

+ 1

(n−1)!

Zt t1

(t−s)ⁿ⁻¹(Λ(u)(s) +q(s))ds for a < t < b,

(2.53)

|u^(j−1)(t)| ≤rj[(t−a)(b−t)]^m−j+1/2 for a < t < b (j = 1, . . . , m), (2.54) u∈Ce_locⁿ⁻¹(]a, b[), and

l→+∞lim u^(j−1)_kl (t) =u^(j−1)(t) (j = 1, . . . , n−1) uniformly in ]a, b[. (2.55) On the other hand, for any t0 ∈]a, b[ and naturall, we have

(t−t0)u⁽ⁿ⁻¹⁾_kl (t) =u⁽ⁿ⁻²⁾_kl (t)−u⁽ⁿ⁻²⁾_kl (t0) + Zt t0

(s−t0)(Λk(uk_l)(s) +qk_l(s))ds. (2.56)

Hence, due to (2.25), (2.47), (2.55), and Lemma 2.6 we get

l→+∞lim u⁽ⁿ⁻¹⁾_kl (t) =u⁽ⁿ⁻¹⁾(t) uniformly in ]a, b[. (2.57) Now it is clear that (2.55), (2.57), and (2.37) results in (2.40) and (2.41). Therefore, u ∈ Ce_loc^{n−1, m}(]a, b[). On the other hand, from (2.53) it is obvious that u is a solution of (1.1). In the case wheren= 2m,from (2.54) equalities (1.2) follow, that is,uis a solution of problem (1.1), (1.2).

Let us show that u is the solution of that problem in the case n = 2m + 1 as well.

In view of (2.54), it suffice to prove that u^(m)(b) = 0. First we find an estimate for the sequence {uk}^+∞k=1. For this, without loss of generality we assume that

t1 ≤t1k (k∈N). (2.58)

From (2.44), by (2.39) and (2.49), it follows the existence of a positive constant ρ0, independent of k, such that

|u^(m+1)_k (t)| ≤

≤ρ0+ 1 (m−1)!

Zt t1

(t−s)^m−1Λk(uk)(s)ds +

Zt t1

(t−s)^m−1qk(s)ds

(2.59)

for t1 ≤t≤t1k,and

||q_k||Le²_{2n−2m−2,2m−2} ≤ρ₀, (2.60) for k ∈N.On the other hand, it is evident that

Zt t1

(t−s)^m−1Λk(uk)(s)ds ≤

Xm j=1

Zt t1

(t−s)^m−1pj(s)u^(j−1)_k (s)ds +

+ Xm

j=1

Zt t1

(t−s)^m−1p_j(s) ^µj(tZ^0k,t1k,s)

s

u^(j)_k (ξ)dξ ds

(2.61)

for t1 ≤t≤t1k (k∈N).

Let, now m >1. From Lemma 2.2 and condition (2.37) we get the estimates Zt

t1

|u^(j)_k (s)|²

(b−s)^2m−2jds ≤

t1k

Z

t0

|u^(j)_k (s)|²

(t_1k−s)^2m−2jds≤2^2mr₀² (2.62) for t1 ≤t≤t1k (j = 1, . . . , m). Then by conditions (2.38) we find

Zt t1

(t−s)^m−1pj(s)u^(j−1)_k (s)ds =

= Zt t1

1 (b−s)^2m−j

∂

∂s

(t−s)^m−1u^(j−1)_k (s) s−a

(b−s)^2m−j Zs t1

(ξ−a)pj(ξ)dξ ds

≤

≤ 4mρj

b−a Z^t

t1

|u^(j−1)_k (s)| (b−s)^m−j+2ds+

Zt t1

|u^(j)_k (s)| (b−s)^m−j+1ds

≤

≤ 4mρj

b−a hZ^t

t1

(u^(j−1)_k (s))²

(b−s)^2m−2j+2ds1/2

+Z^t

t1

(u^(j)_k (s))²

(b−s)^2m−2jds1/2i

×

×Z^t

t1

(b−s)⁻²ds1/2

≤ 2^mmr0ρj

b−a (b−t)^−1/2

(2.63)

for t1 ≤ t ≤ t1k (j = 1, . . . , m). On the other hand, by the Schwartz inequality, the definition of the functions µ_j and (2.4) it is clear that

µj(tZ0k,t1k,s) s

u^(j)_k (ξ)dξ

≤ 2^m−j

(2m−2j−1)!!λj(b, t0k, t1k, s)Z^t1k

t0k

|u^(m)_k (ξ)|²dξ1/2

≤

≤2^mr0λj(b, t0k, t1k, s)

(2.64)

for t1 < s ≤ t1k (j = 1, . . . , m). Then by the integration by parts and (2.38), (2.64) we get

Zt t1

(t−s)^m−1pj(s) ^µj(tZ^0k,t1k,s)

s

u^(j)_k (ξ)dξ ds

≤

≤2^mr0

Zt t1

∂

∂s

(t−s)^m−1 s−a

Z^s

t1

(ξ−a)|pj(ξ)|λj(b, t0k, t1k, ξ)dξ ds

≤2^mr0×

×ρ_j Zt t1

∂

∂s

(t−s)^m−1 s−a

(b−s)^γ1j−m+1/2ds ≤2^mr0ρ_j×

(2.65)